Here, \(\lbrace \beta _t, \eta _t\rbrace \)  are deterministic prespecified upper and lower bound sequences, and \(\chi _t\)  is the gap. We will impose conditions on these sequences later. The particular adaptive schemes designed in [1]}, [2]}, [3]}, [4]} all satisfy the above sandwich condition (e.g., [1]}); and our conditions imposed later on \(\lbrace \beta _t, \eta _t, \chi _t\rbrace \)  are satisfied by those studies as well.
